Diatomic Molecules Atom -> Molecule
Outline 1. Introduction and motivation 1. Overview of natural molecules (and sneak preview) 2. Reasons of interest 2. A Brief History of Molecules 3. Born Oppenheim approximation 4. Examples for electron orbits 1. Linear Combination of Atomic Orbits (LCAO) with H2+ 2. LCAO of Molecular Orbits with H2 3. Short introduction to other Approximations 5. Nuclei vibration and rotation levels 6. Current research 7. Summary 8. Sources
1.1 Overview of natural molecules Mr. Brinclhof (Br I N Cl H O F)
1.2 Reasons of interest LEGO bricks of all molecules e.g. amino acid, sugar, gasoline (chemistry and biology) Mathematical basics for large molecules Experimental tests of theoretical approximations Further possible bindings Basic of complex structures (astrobiophysics)
2 A Brief History of Molecules Etymology: molecule from moles (=mass barrier) ~400 B.C. Platon and Empedocles: Elements: Water, Fire ~350 B.C. Democritus Leucippus : atoms = small particles with properties 1625 Descartes: Resurgence of atomic theory
2 A Brief History of Molecules Etymology: molecule from moles (=mass barrier) ~400 B.C. Platon and Empedocles: Elements: Water, Fire ~350 B.C. Democritus Leucippus : atoms = small particles with properties 1625 1808 1811 Descartes: Resurgence of atomic theory Daltons law of Definite and Multiple Proportions Avogadro: atoms->molecules-> particles
2 A Brief History of Molecules Etymology: molecule from moles (=mass barrier) ~400 B.C. Platon and Empedocles: Elements: Water, Fire ~350 B.C. Democritus Leucippus : atoms = small particles with properties 1625 1808 1811 1903 Descartes: Resurgence of atomic theory Daltons law of Definite and Multiple Proportions Avogadro: atoms->molecules-> particles Gilbert N. Lewis cube modulation
2 A Brief History of Molecules Etymology: molecule from moles (=mass barrier) ~400 B.C. Platon and Empedocles: Elements: Water, Fire ~350 B.C. Democritus Leucippus : atoms = small particles with properties 1625 1808 1811 1903 1916 Descartes: Resurgence of atomic theory Daltons law of Definite and Multiple Proportions Avogadro: atoms->molecules-> particles Gilbert N. Lewis cube modulation Lewis structure -> electron pair bond
2 A Brief History of Molecules Etymology: molecule from moles (=mass barrier) ~400 B.C. Platon and Empedocles: Elements: Water, Fire ~350 B.C. Democritus Leucippus : atoms = small particles with properties 1625 1808 1811 1903 1916 1927 description 2009 Descartes: Resurgence of atomic theory Daltons law of Definite and Multiple Proportions Avogadro: atoms->molecules-> particles Gilbert N. Lewis cube modulation Lewis structure -> electron pair bond Born, Oppenheimer, London, Heitler, Pauling QM IMB first picture of Pentacene with chem. Bonds
2 A Brief History of Molecules Pentacene AFM Atomic Force Microscope IBM picture on molecular level of nano tube using AFM (atomic force microscope
3 Born Oppenheimer Approximation Estimations & principle: No spin Not relativistic: Bohr: => terms negligible for the electrons Nuclei velocity relative to electron movement very slow: Plan: Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Configuration: Hamiltonian SGE:
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Neglect for now! (Electronic SGE)
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Neglect for now! (Electronic SGE) Examples later Now pretend to know a solution : orthonormal & complete set
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Separation: Sum convention New rewritten SGL Multiply by and integrate over r Born Oppenheimer Equation Matrix elements: Non-adiabatic coupling terms (NACT)
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Born Oppenheimer Equation Vibration levels Rotation levels
Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Non adiabatic coupling terms Collapse of the BO approximation Both:
The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only finding the right approximation.» Paul Adrien M. Dirac
To solve: 4.1 LCAO-Linear Combination of Atomic Orbits with H2+ Step overview: A. Neglect nuclei movement B. Estimate a wave function C. Set the distance of nuclei to parameter R D. Normalisation -> interaction integral E. Energy expectation value <H> F. Minimize Energy
4.1 LCAO precision Exact values LCAO values E(R) for the stable symmetric and unstable asymmetri c Overview of effects on the LCAO
Nothing is solved» Me
4.2 Other approximations for H2+ Adjusted wave function +λz: charge distribution drawn out in z- direction -> polarisation +radial probability density η(r). η<1 contraction 1. LCAO simple 2. LCAO contraction η 3. LCAO contraction & polarisation 4. James & Coolidge
4.3 Approximation for H2 LCAO Hamiltonian Solving Idea? Linear combination of atomical Orbits
4.3 Approximation for H2 LCAO Symmetric position, asymmetric spin Antisymmetric position, symmetric spin i) - for two 1-s H-atom functions - for includes two H- atoms ii) - Contains non binding factor -higher energy
4.4 Heitler London Approximation Heitler London
5.Vibration and rotation levels Born Oppenheimer for nuclei & two body problem Rotational symmetry We know the solution
5.Vibration and rotation levels This reduces the equation to New: For small l, depending on l at rl. has a minima Setting leads to an harmonic oscillator
5.Vibration and rotation levels Solving this leads us to: Effective electrical energy Visible light 1 Vibration energy infrared 2 Rotation energy far infrared µwaves 3 1 2 3 ~100nm ~µm ~mm -> cm
6. Current Research - conceptual MPIK nuclear physics early Universe H2 formation Star origin
6. Current Research - conceptual MPIK nuclear physics early Universe H2 formation Star origin, small-particle clouds, fast ion neutral reactions -> molecules
6. Current Research - conceptual MPIK nuclear physics early Universe H2 formation Star origin, small-particle clouds, fast ion neutral reactions -> molecules
6. Current Research - conceptual MPIK nuclear physics early Universe H2 formation Star origin, small-particle clouds, fast ion neutral reactions -> molecules
6. Current Research - conceptual MPIK nuclear physics early Universe H2 formation Star origin, molecular clouds, fast ion neutral reactions -> molecules Atom-molecule dark state from Bose-Einstein-Condensate BEC ( a>) + Laser beam -> frees Rb atoms to Rb2 molecules (photoassociation) Second laser beam: certain frequency ->suppression of photoassociation Dark state ~ c1 g> - c2 a>
7. Summary Natural diatomic molecules Br I N Cl H O F Born Oppenheimer LCAO (H2+) Heitler London Vibration & rotation levels Current research H2 formation main coolant, Rb superposition
Thank you for your Attention!
Introduction - History - Born Oppenheimer - Electron orbits - Vibration and rotation Summary - Sources 8. References Literature Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 (6 April 1929) Schwabl, Franz (2002). Quantenmechanik. 6. Auflage, Berlin: Springer Verlag Born, Max & Oppenheimer, Robert (1927), Analen der Physik Zur Quantenheorie der Molekeln. Vierte Folge Band 84, Leipzig: Verlag von Joh. Ambrosius Barth Lewis, Gilbert. N., (April 1916), The Atom and the Molecule. Boston. Verlag unklar Demtröder, Wolfgang, (Februar 2005), Experimentalphysik 3- Atome Moleküle und Festkörper. Auflage 3, New York: Springer Verlag Lector of Jack Simpson, PhD in Toronto, Canada Area, Lecture to Born and Oppenheimer Approximation History http://en.wikipedia.org/wiki/history_of_molecular_theory Gilbert N. Lewis Original Work: http://osulibrary.oregonstate.edu/specialcollections/coll/pauling/bond/papers/corr216.3-lewispub-19160400-07.html BBC News Penatacine AFM Image http://news.bbc.co.uk/2/hi/8225491.stm Born Oppenheimer & solutions Technische Universität Braunschweig: http://www.pci.tu-bs.de/aggericke/lehre/quantenmechanik/gdanitz/node4.html University of Oxford Department of Physics: http://www2.physics.ox.ac.uk/sites/default/files/2011-09-16/born_oppenheimer_pdf_31916.pdf New York University: http://www.nyu.edu/classes/tuckerman/quant.mech/lectures/lecture_10/node1.html Universität Jena: http://www.theochem.uni-jena.de/teaching/ws1112/vf/vl06.pdf Paul Dirac Quote: http://en.wikiquote.org/wiki/paul_dirac Current research MPIK HEIDELBERG & http://www.hkreckel.de/molecular_hydrogen.html Ultra Cold Gases and Molekular Superposition University of Innsbruck: http://www.uibk.ac.at/exphys/ultracold/?http://www.uibk.ac.at/exphys/ultracold/projects/rubidium/dark
Outtakes
Introduction - History - Born Oppenheimer - Electron orbits - Vibration and rotation - Summary Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Justification of neglecting the first NACT term insignifican t
Introduction - History - Born Oppenheimer - Electron orbits - Vibration and rotation - Summary Hamiltonian Neglect nuclei velocity Solve electron wave equa. Separation & rewrite Born Oppenheimer Negelct Matrix elements Justification for the second NACT term Estimation with oscillation wave function: Let be a typical nucleus amplitude terms of size insignifica nt